Yes. (Is it significant that you used an uppercase and a lowercase a?) The magnitude of a vector is a scalar, and a vector multiplied or divided by a scalar is a vector.

This is called the triangle inequality.My other question is, using vectors explain why |a+b| is always less than |a|+|b|.

I think it is always smaller than |a|+|b| because it goes in a straight line, from point a to point c, and not a-to-b-to-c. For example, |a+b| of the 3,4,5 triangle is 5, while |a|+|b| is 7. This means you will have a less magnitude because you will be going in a straight line.